Problem: Factor the following expression: $-3$ $x^2$ $-8$ $x+$ $35$
Solution: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-3)}{(35)} &=& -105 \\ {a} + {b} &=& & & {-8} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-105$ and add them together. Remember, since $-105$ is negative, one of the factors must be negative. The factors that add up to ${-8}$ will be your ${a}$ and ${b}$ When ${a}$ is ${7}$ and ${b}$ is ${-15}$ $ \begin{eqnarray} {ab} &=& ({7})({-15}) &=& -105 \\ {a} + {b} &=& {7} + {-15} &=& -8 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-3}x^2 +{7}x {-15}x +{35} $ Group the terms so that there is a common factor in each group: $ ({-3}x^2 +{7}x) + ({-15}x +{35}) $ Factor out the common factors: $ x(-3x + 7) + 5(-3x + 7) $ Notice how $(-3x + 7)$ has become a common factor. Factor this out to find the answer. $(-3x + 7)(x + 5)$